Calculation
F₁ = F₂ · e^(μ·θ)
The tension ratio grows exponentially with friction coefficient and wrap angle (Euler-Eytelwein equation).
F₁/F₂ = e^(μθ)
Belt tension ratio using the capstan equation
The tension ratio grows exponentially with friction coefficient and wrap angle (Euler-Eytelwein equation).
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Tight-side tension F₁ — Force in the tight (driving) belt strand.
- Slack-side tension F₂ — Force in the slack belt strand.
- Friction coefficient μ — Sliding friction coefficient between belt and pulley.
- Wrap angle θ — Wrap angle at the pulley considered.
Example
μ = 0.3 and θ = 180° (π rad) give a maximum tight-side tension of about 256.7 N at F₂ = 100 N.
Assumptions and limits
At the point of impending slip; centrifugal effects and belt stiffness are excluded.
Understand Belt tension ratio using the capstan equation
This calculator finds the missing quantity among tight-side tension F₁, slack-side tension F₂, friction coefficient μ and wrap angle θ from the other two, using the capstan (Euler-Eytelwein) equation. It answers the central question of any belt drive design: how much more tension can the tight side of a belt carry than the slack side before the belt slips?
What does this quantity describe?
At the point of impending slip, the ratio of tight-side to slack-side tension relates exponentially to friction coefficient and wrap angle: F₁ = F₂·e^(μθ), with θ in radians. This relationship, known as the capstan or Euler-Eytelwein equation, applies generally to any rope or belt sliding over a cylinder or pulley.
A sailor can control a heavy sail with little hand force by wrapping the line several times around a cleat: each extra turn increases the angle θ and so exponentially raises the force the holding hand can resist against the sail's load. The exact same capstan equation determines, for a belt drive, how much tension difference is possible between the two belt strands before the belt slips on the pulley.
Formula and variables
F₁ = F₂ · e^(μ·θ)
F₁ = F₂ · e^(μθ)F₂ = F₁ · e^(−μθ)μ = ln(F₁/F₂) / θθ = ln(F₁/F₂) / μ
| Symbol / input | Meaning |
|---|---|
| Tight-side tension F₁ | Force in the tight (driving) belt strand. |
| Slack-side tension F₂ | Force in the slack belt strand. |
| Friction coefficient μ | Sliding friction coefficient between belt and pulley. |
| Wrap angle θ | Wrap angle at the pulley considered. |
Choose the inputs correctly
Enter two of the four quantities. Use the wrap angle at the pulley where slipping would occur first for θ — usually the smaller pulley of a belt drive.
How to use the calculator
Select the target quantity and enter the other known values with units. The result describes the limiting case at the point of impending slip; for safe operation, the actual tension difference should keep a safety margin below this limit.
Worked example
Given F₂ = 100 N, μ = 0.3 and θ = 180° (π radians), substitution gives F₁ = 100 · e^(0.3·π) ≈ 256.6 N.
Understand the result and units
A maximum tight-side tension of 256.6 N at 100 N slack-side tension shows that at most a tension difference of about 156.6 N is usable under these conditions before the belt slips. This difference multiplied by pulley radius gives the maximum transmittable torque.
Forces in newtons (N); friction coefficient μ is dimensionless; wrap angle is given in degrees or radians — radians are always required for the calculation itself.
Typical applications
The capstan equation is used to design belt and rope drives, conveyor drives, winches, and to size brake bands and rope clamps.
Assumptions, limits and common mistakes
The formula describes the static limiting case immediately before slip for a massless, inextensible belt. At high belt speeds, centrifugal force reduces effective contact pressure, lowering the actually transmittable force difference below this ideal value.
Common mistake: Do not substitute wrap angle in degrees into the formula without converting to radians first — the calculator handles this conversion automatically, but a manual calculation must use θ in radians. Also, do not equate friction coefficient μ with the static friction of a dry, clean surface if the belt is actually soiled, worn or oiled.
Frequently asked questions
Why does wrapping a line a few times around a cleat hold such large loads?
Because the sustainable tension difference grows exponentially with wrap angle. Each additional turn adds 360° (2π) to θ, multiplying the possible tension difference.
Which wrap angle matters for a belt drive?
The smaller of the two angles, usually at the small pulley, since slipping starts there first.
How do I get transmittable torque from this result?
Multiply the difference between tight- and slack-side tension (F₁ − F₂) by the radius of the pulley considered, for example using the existing torque calculator.